# Linear Algebra with Applications (Classic Version), 5th edition

Published by Pearson (March 15, 2018) © 2019

**Otto Bretscher**

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This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price.

### Emphasizes linear transformations as a unifying theme

**Linear Algebra with Applications** offers the most geometric presentation available. This elegant textbook combines a user-friendly presentation with straightforward, lucid language to clarify and organize techniques and applications. Exercises and examples make up the heart of the text, with abstract exposition kept to a minimum. Exercise sets are broad and varied and reflect the author's creativity and passion for this course. The **5th Edition** reflects careful review and appropriate edits throughout.

### Hallmark features of this title

**Linear transformations**are introduced early to make the discussion of matrix operations more meaningful and easier to navigate.**Visualization and geometrical interpretation**are emphasized extensively throughout.**Abundant problems, exercises and applications**help students assess their understanding and master the material.**Abstract concepts are****introduced gradually**. Major ideas are carefully developed at various levels of generality before abstract vector spaces are introduced.**Discrete and continuous dynamical systems**are used as a motivation for eigenvectors and as a unifying theme thereafter.**50 - 60 True/False questions**conclude every chapter, testing conceptual understanding and encouraging students to read the text.

### New and updated features of this title

**A large number of exercises**have been added to the problem sets, from the elementary to the challenging and from the abstract to the applied.- For example, there are quite a few new exercises on Fibonacci Matrices and their eigenvectors and eigenvalues.

- Throughout the text, the author added an ongoing discussion of the
**mathematical principles behind search engines**(and the notion of PageRank in particular) with dozens of examples and exercises.- Besides being an interesting and important contemporary application of linear algebra, this topic allows for an early and meaningful introduction to dynamical systems, one of the main themes of this text, naturally leading up to a discussion of diagonalization and eigenvectors.

- A
**new appendix**offers a brief discussion of the proof techniques of Induction and Contraposition. **Hundreds of improvements**include offering hints in a challenge problem, for example, or choosing a more sensible notation in a definition.

**1. Linear Equations**

1.1 Introduction to Linear Systems

1.2 Matrices, Vectors, and Gauss-Jordan Elimination

1.3 On the Solutions of Linear Systems; Matrix Algebra

**2. Linear Transformations**

2.1 Introduction to Linear Transformations and Their Inverses

2.2 Linear Transformations in Geometry

2.3 Matrix Products

2.4 The Inverse of a Linear Transformation

**3. Subspaces of R^{n} and Their Dimensions**

3.1 Image and Kernel of a Linear Transformation

3.2 Subspace of *R ^{n}*; Bases and Linear Independence

3.3 The Dimension of a Subspace of *R ^{n}*

3.4 Coordinates

**4. Linear Spaces**

4.1 Introduction to Linear Spaces

4.2 Linear Transformations and Isomorphisms

4.3 The Matrix of a Linear Transformation

**5. Orthogonality and Least Squares**

5.1 Orthogonal Projections and Orthonormal Bases

5.2 Gram-Schmidt Process and *QR* Factorization

5.3 Orthogonal Transformations and Orthogonal Matrices

5.4 Least Squares and Data Fitting

5.5 Inner Product Spaces

**6. Determinants**

6.1 Introduction to Determinants

6.2 Properties of the Determinant

6.3 Geometrical Interpretations of the Determinant; Cramer's Rule

**7. Eigenvalues and Eigenvectors**

7.1 Diagonalization

7.2 Finding the Eigenvalues of a Matrix

7.3 Finding the Eigenvectors of a Matrix

7.4 More on Dynamical Systems

7.5 Complex Eigenvalues

7.6 Stability

**8. Symmetric Matrices and Quadratic Forms**

8.1 Symmetric Matrices

8.2 Quadratic Forms

8.3 Singular Values

**9. Linear Differential Equations**

9.1 An Introduction to Continuous Dynamical Systems

9.2 The Complex Case: Euler's Formula

9.3 Linear Differential Operators and Linear Differential Equations

Appendix A. Vectors

Appendix B: Techniques of Proof

Answers to Odd-numbered Exercises

Subject Index

Name Index

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